On the structure of factor Lie algebras
Bull. Korean Math. Soc. 2017 Vol. 54, No. 2, 455-461
Published online March 9, 2017
Printed March 31, 2017
Homayoon Arabyani, Farhad Panbehkar, and Hesam Safa
Neyshabur Branch, Islamic Azad University, Neyshabur Branch, Islamic Azad University, University of Bojnord
Abstract : The Lie algebra analogue of Schur's result which is proved by Moneyhun in 1994, states that if $L$ is a Lie algebra such that $ \mathrm{dim}L/Z(L)=n$, then $ \mathrm{dim}L_{(2)}=\frac{1}{2}n(n-1)-s$ for some non-negative integer $s$. In the present paper, we determine the structure of central factor (for $s=0$) and the factor Lie algebra $L/Z_2(L)$ (for all $s\geq0$) of a finite dimensional nilpotent Lie algebra $L$, with $n$-dimensional central factor. Furthermore, by using the concept of $n$-isoclinism, we discuss an upper bound for the dimension of $L/Z_n(L)$ in terms of dim$L_{(n+1)}$, when the factor Lie algebra $L/Z_n(L)$ is finitely generated and $n\geq 1$.
Keywords : factor Lie algebra, $n$-isoclinism, nilpotent Lie algebra
MSC numbers : Primary 17B30, 17B60; Secondary 17B99
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